Integrable natural systems in pseudoeuclidean plane

The description of all natural Hamiltonian systems in pseudoeuclidean plane is given which have an additional integral, quadratic in momenta. For polynomial potentials the recurrent formula generating bi-Hamiltonian chains is obtained. The Darboux—Nijenhuis coordinates are introduced due to which dynamic equations are integrated in quadratures. The examples of bi-Hamiltonian chains are considered.